Treatment vs. effect contrasts

Hi Wolfgang and Michel, here is a simple example to show the differences between treatment and effect contrasts.

Example dataset

Create a random dataset with criteria y, predictors x1, x2 and gender.

All variables are correlated.

set.seed(1234)
n <- 10000
gender <- rep(0:1, each = n/2)
y <- sample(0:10, n, replace = TRUE) + gender * sample(0:10, n, replace = TRUE)
x1 <- sample(0:10, n, replace = TRUE) + gender * y
x2 <- x1 + sample(0:10, n, replace = TRUE) + gender * y

y <- y + (x1 > median(x1) & x2 > median(x2) & gender == 1) * sample(0:10, n, replace = TRUE)

dat <- data.frame(y = y, x1 = x1, x2 = x2, gender = gender)

Descriptives

psych::corr.test(dat)

Call:psych::corr.test(x = dat)
Correlation matrix 
          y   x1   x2 gender
y      1.00 0.79 0.82   0.66
x1     0.79 1.00 0.94   0.74
x2     0.82 0.94 1.00   0.79
gender 0.66 0.74 0.79   1.00
Sample Size 
[1] 10000
Probability values (Entries above the diagonal are adjusted for multiple tests.) 
       y x1 x2 gender
y      0  0  0      0
x1     0  0  0      0
x2     0  0  0      0
gender 0  0  0      0

 To see confidence intervals of the correlations, print with the short=FALSE option

psych::describe(dat)

	vars	n	mean	sd	median	trimmed	mad	min	max	range	skew	kurtosis	se
y	1	10000	9.5544	6.773613	8.0	9.044000	7.4130	0	30	30	0.6076269	-0.5395695	0.0677361
x1	2	10000	10.0063	6.672279	9.0	9.585875	7.4130	0	30	30	0.5162774	-0.5350028	0.0667228
x2	3	10000	19.9998	12.575874	16.0	19.046000	11.8608	0	58	58	0.5910003	-0.6639372	0.1257587
gender	4	10000	0.5000	0.500025	0.5	0.500000	0.7413	0	1	1	0.0000000	-2.0002000	0.0050003

Contrasts

The left part of the table is with gender as treatment contrast (0 vs. 1) and the right part with gender as effect contrast (-1 vs. 1)

# Gender has values 0 vs. 1 (treatment contrast)
fit1 <- lm(y ~ gender * x1 * x2, data = dat)

# Gender hast -1 vs. 1 (effect contrast)
dat$gender <- car::recode(dat$gender, "0 = -1; 1 = 1")

fit2 <- lm(y ~ gender * x1 * x2, data = dat)

sjPlot::tab_model(fit1, fit2, show.se = TRUE, show.ci = FALSE, col.order = c("est", "se", "std.est", "p"), digits = 4)

	y			y
Predictors	Estimates	std. Error	p	Estimates	std. Error	p
(Intercept)	5.1849	0.1890	<0.001	-0.8279	0.1892	<0.001
gender	-12.0257	0.3784	<0.001	-6.0128	0.1892	<0.001
x1	0.0226	0.0422	0.592	0.3045	0.0261	<0.001
x2	-0.0066	0.0240	0.782	0.3072	0.0145	<0.001
gender × x1	0.5637	0.0521	<0.001	0.2819	0.0261	<0.001
gender × x2	0.6277	0.0289	<0.001	0.3139	0.0145	<0.001
x1 × x2	-0.0026	0.0036	0.482	-0.0078	0.0019	<0.001
(gender × x1) × x2	-0.0105	0.0037	0.005	-0.0052	0.0019	0.005
Observations	10000			10000
R2 / R2 adjusted	0.745 / 0.745			0.745 / 0.745